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Author Topic: True 3D mandelbrot type fractal  (Read 263127 times)
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Daniel_P
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« Reply #450 on: November 13, 2009, 02:30:28 PM »

So, have New Scientist, Scientific American, etc been in touch yet?

I'm guessing we might see some of these images on more than a few magazine covers in the near future.
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cKleinhuis
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« Reply #451 on: November 13, 2009, 03:04:56 PM »

@daniel_p i think there should be at least one scientific description of what we they are doing here, it would be a pleasure
for me to collect all the informations we have here, but it would take at least 2 or 3 monthes ( hello bachelor thesis, actually i am trying to find someone who would find that interesting )

cheers  afro angel police sad
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fpsunflower
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« Reply #452 on: November 13, 2009, 05:05:25 PM »

Thanks, I read the Wikipedia article, but I still do not see how this gives us the normal vector in a point, "for free".

Another good explanation that describes duals in more concrete coding terms: http://homepage.mac.com/sigfpe/paper.pdf

Not quite "free", of course you need to implement your dual number class and define add, mult, sqrt, log, div, etc ...



iq: thanks for that explanation, but I also got a bit lost in those final steps about the exact leading term.
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bib
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« Reply #453 on: November 13, 2009, 05:14:28 PM »


Bib, I'm not sure what to make of that latest blue image - maybe render a different angle and zoom in. It could be very cool...

I try to. But navigating using the parameters variations instead of "natural" 2D zooming in UF is quite tough.

I know there is nothing revolutionnary in the images I post in this thread compared to all the ground-breaking innovations we can see here, but it's just to show that zooming into 3D fractals that look like whipped cream at first sight can reveal interesting spirals and so on...
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David Makin
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« Reply #454 on: November 13, 2009, 10:43:53 PM »

The formula comes, more or less, from this sequence of steps: (assume we have the regular quadratic case)
<snip>

That's interesting, I tried my own derivation of a DE formula and got it working based on the distance to the next iteration boundary.
Unfortunately I did it on scrap paper, tried several renders based on a version where the next iteration boundary was when the final magnitude on the bailout iteration became (actual final magnitude)^(1/n) where n is the divergence then realised a better method would be to actually use the smooth iteration value instead and base it on when the smooth iteration value increases by 1 smiley
Anyway it worked and when I derived the smooth iteration version I basically arrived at the normal DE formula.
All I have left of my derivations is this note on a scrap of paper I had left:

"Iteratively solve the Newton for lambda where lambda is the distance aliong the ray for intersection with the bailout sphere."

i.e. iterate both the main formula and the Newton at the same time.

Here's what I got when I used the final magnitude derivation on a Quaternion:

http://makinmagic.deviantart.com/art/Improved-Quat-Minibrot-127157826

(see the comments)


« Last Edit: November 13, 2009, 11:18:26 PM by David Makin » Logged

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David Makin
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« Reply #455 on: November 13, 2009, 11:37:41 PM »

Just a note to anyone reading the thread who may be unfamilier with |z| as used in standard maths notation and/or as used in Fractint and Ultra Fractal.

If z is complex as (x+iy) then in standard maths notation |z| = sqrt(x^2 + y^2)

BUT in Ultra Fractal and Fractint then |z| = x^2 + y^2 and cabs(z) is used as sqrt(x^2 + y^2) (cabs for complex absolute).

I was prompted to mention this after IQ's query regarding the correct scaling of the DE value, I also couldn't work out how "0.5" was arrived at. In UF and Fractint:

0.5*log(|z|) is equivalent to log(|z|) in standard math notation smiley

And at one point I considered that maybe the 0.5 came from mistranslations between fractal program and standard math notations.

But I don't think the 0.5 comes from that, I think it really comes from the fact that the value calculated is only an *estimate* and the 0.5 is simply used to avoid cases where the estimate is larger than the true distance - but that's just my opinion smiley
In fact the DE value I use for the Normals is assumed to be correct without the "0.5", I do however multiply the DE value by 0.5 before using the value as a step distance during ray stepping.
Also I should mention that (based on how I had to scale the DE values used to compute the normals to get correct lighting) I found (experimentally) that the visibly correct (Ultra Fractal) final DE calculation was:

            dist = 0.5*sqrt(@mpwr-1.0)*log(magn)*sqrt(magn/|dzri|)

for a Julibrot where @mpwr is the power in z^power+c, magn is |final z| (i.e. x^2+y^2) and dzri is the final derivative value.
However looking at IQ's derivation I suspect that:

            dist = 0.25*@mpwr*log(magn)*sqrt(magn/|dzri|)

May be more accurate smiley Edit: Just tried it and it doesn't seem to be better, in fact it definitely overestimates at higher powers - I not only got gradually darker and darker shading but started getting gross overstepping as well when the method using sqrt(@mpwr-1) still worked fine.
Just to add I actually checked it on quaternions and the White/Nylander.

« Last Edit: November 14, 2009, 02:25:22 AM by David Makin » Logged

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bugman
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« Reply #456 on: November 14, 2009, 05:11:39 AM »

I thoroughly double checked my non-trigonometric formulas again, and I found that the formula for {x,y,z}^4 had a minus sign error. But all the other formulas came out correct. I went back and made the correction to the original post here:
http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg8426/#msg8426
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David Makin
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« Reply #457 on: November 14, 2009, 05:30:21 AM »

I thoroughly double checked my non-trigonometric formulas again, and I found that the formula for {x,y,z}^4 had a minus sign error. But all the other formulas came out correct. I went back and made the correction to the original post here:
http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg8426/#msg8426

I'm not sure what went wrong with the z^3+c when I tried it (or the other higher powers where you use abs and sign) maybe I'm missing something obvious.
Anyway the formula I quoted here http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/msg8475/#msg8475 can be expanded to use reals for a given power and then optimised - that formula definitely exactly matches the trig version (at least for integer powers from 2 to 60, I haven't checked fractional or negative powers).
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David Makin
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« Reply #458 on: November 14, 2009, 06:29:29 AM »

Hi all,

I just updated the WIP3D formula, new link:

http://www.fractalgallery.co.uk/MMFWip3D.zip

Again unzip, copy all the text and paste into an open UF fractal window (also requires UF5).
This time the UPR is an example render of Pickover stalks (orbit trap) - note that "auto distances" is disabled and the solid threshold is much larger than used for solid on iteration/distance estimate.
Several more options in the formula plus some bugfixes in the analytical distance estimate methods.
This is actually named as a different formula so as not to cause compatibility issues with any renders you've already done smiley
« Last Edit: November 14, 2009, 06:32:14 AM by David Makin » Logged

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JosLeys
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« Reply #459 on: November 14, 2009, 11:58:58 AM »

Here is a very simple general procedure to obtain sin(p.a) and cos(p.a) if you know the value for sin(a) and cos(a) :

Let S=sin(a) and C=cos(a) and make a complex number z=S+iC, and put z1=z

Do z1=z1.z,   now real(z1)=cos(2a) and imag(z1)=sin(2a)  (z1=C^2-S^2 +i.2.S.C)

Do again z1=z1.z now real(z1)=cos(3a) and imag(z1)=sin(3a)

(because (sin(2a)+i.cos(2a))(sin(a)+i.cos(a) gives as real part sin(2a)cos(a)+cos(2a)sin(a)=sin(3a) and as imaginary part cos(2a)cos(a)-sin(2a)sin(a)=cos(3a) )

...
..(the (p-1)-th time)  real(z1)=cos(pa) and imag(z1)=sin(pa)

So no need for very long complicated formulas if you want do to a non-trig p-th power...
The drawback is that you need sin(a) and cos(a) which involves a square root.

For even powers this can be adapted so that you only need (sin(a))^2 and (cos(a))^2, which does away with the square root. However, when using a distance estimate based on the derivative, one always needs also the value for sin((p-1)a) and cos((p-1)a), so getting a value for sin(a) and cos(a) cannot be avoided then.

Also, if you want to do powers<1, it is no good of course.
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Aexion
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« Reply #460 on: November 14, 2009, 06:20:56 PM »

Wow, I'm away for a couple of days, and then I miss all this!

Aexion, you've got to one of my fave fractal artists ever - I love your recent "The Hexahedral Puzzle" - it has a lovely Aztec feel about it. Still think your Sunset Castle (and Return thereof) is awesome.


Thanks a lot Twinbee!!  smiley
It's nice to be here, and BTW, your renders are absolutely amazing!
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ker2x
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« Reply #461 on: November 14, 2009, 06:53:49 PM »

Hi all,

I just updated the WIP3D formula, new link:

http://www.fractalgallery.co.uk/MMFWip3D.zip

Again unzip, copy all the text and paste into an open UF fractal window (also requires UF5).
This time the UPR is an example render of Pickover stalks (orbit trap) - note that "auto distances" is disabled and the solid threshold is much larger than used for solid on iteration/distance estimate.
Several more options in the formula plus some bugfixes in the analytical distance estimate methods.
This is actually named as a different formula so as not to cause compatibility issues with any renders you've already done smiley


Hi ! Thank you for sharing this.
I'm playing with incendia (which is awesome) since yesterday.

And with UF5 (demo Grin with closed eyes) and your formula.
Sadly, not good enough to understand the math behind (*sigh*) i just click here and there and found that i could do some interestring rendering with the "Truly 3D Mandelbrot". i don't know how "true" is my rendering and how it could be even close to anything related to a mandelbrot fractal but... it's nice.

I've seen the pic of "3D Mandelbrot" seen from the inside... it's truely awesome. But it look like most of 3D Mandelbrot rendering are done with custom renderer and not distributed at all... am i correct ?

Thx again !!

PS about the image below. "Truely 3D mandelbrot" after randomly playing with parameters

Edit : i finally found how to adjust parameters to have a 3D Mandelbrot, adjust accuracy and detail.
It is, mostly, all about "Solid Thresold" to be able to dive into details smiley


* keru3D-2.jpg (62.78 KB, 800x800 - viewed 1726 times.)
« Last Edit: November 14, 2009, 07:32:04 PM by ker2x » Logged

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bib
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« Reply #462 on: November 14, 2009, 08:55:46 PM »

Some vids based on the old M-true3D formula. Nice to watch but quite disappointing indeed sad
Watch the first one til the end you'll recognize the 2D M-set shape.
In the second one I tried to use some rotation to enhance the 3D impression
<a href="http://www.youtube.com/v/mIbDAunza6Q&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/mIbDAunza6Q&rel=1&fs=1&hd=1</a> <a href="http://www.youtube.com/v/uDUWEBAk8Ys&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/uDUWEBAk8Ys&rel=1&fs=1&hd=1</a>
« Last Edit: November 14, 2009, 09:00:58 PM by bib » Logged

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JosLeys
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« Reply #463 on: November 14, 2009, 11:11:04 PM »

On the question which factor should precede the analytical distance estimate formula :

is it
Quote
dist = 0.5*sqrt(@mpwr-1.0)*log(magn)*sqrt(magn/|dzri|)
or
Quote
dist = 0.25*@mpwr*log(magn)*sqrt(magn/|dzri|)
?

Let's change notation first : for the spherical case for any power I use DE=log(R)*sqrt(R)/(dR)/2, where dR is the derivative. So my factor is just 0.5.

To test how accurate this is, I took a Julia with seed (0,0,0) which produces a sphere of radius 1, so we now know the real distance for any point on a ray.

I'm printing the results below for a point progressing from a distance 2 from the surface of the sphere. The first column is the number of iterations, the second is the calculated DE, and the third is the real distance. As the point is still "far" away, and the number of iterations to reach bailout is low, then DE is exaggerated. (that's why an extra factor to temper DE is always needed). However, as the point gets closer, DE becomes very accurate.
The numbers are from a power 8 Julia, but I see almost exactly the same numbers for a degree 2.

power=8
start
         DE                        real distance
1 , 3.29583686625158 , 1.99999999999248
1 , 0.407849058786746 , 0.352081566866694
2 , 0.158627173103063 , 0.148157037473318
2 , 0.0711605799066145 , 0.0688434509217899
2 , 0.033810346387068 , 0.033263160968481
3 , 0.01649105643562 , 0.0163579877749456
3 , 0.00814527732540994 , 0.00811245955713247
3 , 0.00404797039654616 , 0.00403982089442678
4 , 0.00201786652103339 , 0.00201583569615593
4 , 0.00100740958383335 , 0.00100690243564117
4 , 0.000503324618422875 , 0.000503197643723752
5 , 0.000251567358870418 , 0.000251535334513164
5 , 0.000125759953482483 , 0.000125751655076556
5 , 6.28740467193265E-005 , 6.28716783381833E-005
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Buddhi
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« Reply #464 on: November 15, 2009, 04:59:28 PM »

Hi
I'm still fighting with some kind of volumetric rendering. Now it's not fixed grid. Step depends on iteration count, perspective factor (bigger step in bigger distances). Additionally for improve performance and quality I use binary searching algorithm. I think it's quite enough for this kind rendering.
Generally I have lot's of trouble to render that deep perspective view, because for acceptable performance I have to add lots of factors to each shading algorithm. Details in background are 1000 times bigger than in foreground. For the farthest elements quality of shaders are much less accurate. I made the same for more transparent details. Without this optimisation this fractal will be rendered approximately in one month. Now it is possible in 8 hours with global illumination.

Render parameters:
resolution: 1280x1280
camera target: {-1.7844592849042, 0, 0}
zoom: 10000x
max. number of iterations: 80
fog range: between 65 and 80 iteration


http://www.fractalforums.com/gallery/?sa=view;id=1058
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